Semidualizing and tilting adjoint pairs, applications to comodules. (Q2017707)

Semidualizing \(R\)-modules over an associative unitary ring \(R\) were studied in the literature by several authors under different names, see for example \textit{E. S. Golod} [Tr. Mat. Inst. Steklova 165, 62-66 (1984; Zbl 0577.13008)], \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267-284 (1973; Zbl 0272.13009)], and the monograph by \textit{W. V. Vasconcelos} [Divisor theory in module categories. North-Holland Mathematics Studies. 14. Notas de Matematica. 5. Amsterdam-Oxford: North-Holland Publishing Comp. (1974; Zbl 0296.13005)]. In the paper, a unification of the concept of semidualizing object is presented by extending it to the framework of adjoint functors. This is done by introducing the concept of semidualizing adjoint pair and a tilting adjoint pair. The authors relate semidualizing adjoint pairs and tilting adjoint pairs with a generalization of the well-known Brenner-Butler Theorem. This general theory is successfully applied to the case of comodule categories over \(K\)-coalgebras, where \(K\) is a field. In this case the concept of semidualizing bicomodule is introduced. It is proved that if \(C\) and \(C'\) are one-sided semiperfect \(K\)-coalgebras that admit a semidualizing bicomodule of finite injective dimension then: (i) the Grothendieck groups \(K_0(C)\) and \(K_0(C')\) are isomorphic, and (ii) \(\dim_KC\) is finite if and only if \(\dim_KC'\) is finite. A relation between the concepts of semidualizing comodule and \(f\)-cotilting \(C\)-comodule in the sense of \textit{D. Simson} [Arab. J. Sci. Eng., Sect. C, Theme Issues 33, No. 2, 421-445 (2008; Zbl 1186.16039)] is given. In particular, it is proved that the dual of any \(f\)-cotilting \(C\)-comodule over a right semiperfect coalgebra \(C\) is a tilting module in the sense of \textit{Y. Miyashita} [Math. Z. 193, 113-146 (1986; Zbl 0578.16015)].